Robust Extension of the Multivariate Fay-Herriot Model for Data with Outliers

Abstract Outliers commonly arise in real situations, particularly in small areas where outliers could occur due to model errors or survey errors. In this study, we investigate a robust approach for handling outliers in a multivariate small area model, where there are multiple response variables of interest. Specifically, we consider the multivariate Fay-Herriot (MFH) model, which commonly assumes multivariate normal distributions for area random effects and sampling errors. However, it is known that the normality assumption is usually sensitive to outliers or data with heavy tails. Therefore, we propose the multivariate t-distribution as the alternative underlying distribution, called the Mt-FH model. We also construct parameter estimation methods for the model by considering generalizations of the likelihood function and the EM algorithm to enhance the accuracy and avoid computational issues. In particular, we propose two parameter estimation methods for the model based on the (profile) Lq-likelihood approach and the expectation conditional maximization (ECM) algorithm, called the MLq-ECM and profile MLq-ECM. We conduct a simulation study to investigate the efficiency of the Mt-FH model and the (profile) MLq-ECM methods in handling outliers and deviations of the data from the normality assumption. The outlier patterns considered in our study are the mean shift and variance shift patterns. The simulation results based on biases and mean squared errors suggest that the Mt-FH models can improve the MFH model in the presence of outliers. Finally, we assess the performance of the Mt-FH model through an analysis of the welfare variables in Thailand. The results based on the AIC and the percentage reduction of the MSE estimator highlight the advantages of the Mt-FH model over the MFH model. Furthermore, the integration of the MLq-ECM approach further strengthens the performance of the Mt-FH model.